Topological Degree Approach to Bifurcation Problems by Michal FeckanTopological Degree Approach to Bifurcation Problems by Michal Feckan

Topological Degree Approach to Bifurcation Problems

byMichal Feckan

Paperback | November 30, 2010

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This book is devoted to bifurcations of periodic, subharmonic and chaotic oscillations, and travelling waves in nonlinear differential equations and discrete dynamical systems by using the topological degree theory both for single-valued and multi-valued mappings in Banach spaces. Original bifurcation results are proved with applications to a broad variety of nonlinear problems ranging from non-smooth and discontinuous mechanical systems, weakly coupled oscillators, systems with relay hysteresis, through infinite chains of differential equations on lattices involving also spatially discretized partial differential equations, and to string and beam partial differential equations. Next, the chaotic behaviour is also investigated for maps possessing topologically transversally intersecting invariant manifolds. Moreover, periodic orbits with arbitrarily high periods, the so-called blue sky catastrophe, are shown for reversible differential systems and maps. Finally, bifurcations of large amplitude oscillations for discontinuous undamped wave partial differential equations are given as well.This book is mainly intended for post-graduate students and researchers in mathematics with an interest in applications of topological bifurcation methods to dynamical systems and nonlinear analysis, in particular to differential equations and inclusions, and maps. But, among others, it could also be used either by physicists studying oscillations of nonlinear mechanical systems or by engineers investigating vibrations of strings and beams, and electrical circuits.
Title:Topological Degree Approach to Bifurcation ProblemsFormat:PaperbackDimensions:270 pages, 9.25 × 6.1 × 0.68 inPublished:November 30, 2010Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048179696

ISBN - 13:9789048179695

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Table of Contents

1. Introduction 1.1. Preface 1.2. An Illustrative Perturbed Problem 1.3. A Brief Summary of the Book 2. Theoretical Background 2.1. Linear Functional Analysis 2.2. Nonlinear Functional Analysis 2.2.1. Implicit Function Theorem 2.2.2. Lyapunov-Schmidt Method 2.2.3. Leray-Schauder Degree 2.3. Differential Topology 2.3.1. Differentiable Manifolds 2.3.2. Symplectic Surfaces 2.3.3. Intersection Numbers of Manifolds 2.3.4. Brouwer Degree on Manifolds 2.3.5. Vector Bundles 2.3.6. Euler Characteristic 2.4. Multivalued Mappings 2.4.1. Upper Semicontinuity 2.4.2. Measurable Selections 2.4.3. Degree Theory for Set-Valued Maps 2.5. Dynamical Systems 2.5.1. Exponential Dichotomies 2.5.2. Chaos in Discrete Dynamical Systems 2.5.3. Periodic O.D.Eqns 2.5.4. Vector Fields 2.6. Center Manifolds For Infinite Dimensions 3. Bifurcation of Periodic Solutions 3.1. Bifurcation of Periodics from Homoclinics I 3.1.1. Discontinuous O.D.Eqns 3.1.2. The Linearized Equation 3.1.3. Subharmonics for Regular Periodic Perturbations 3.1.4. Subharmonics for Singular Periodic Perturbations 3.1.5. Subharmonics for Regular Autonomous Perturbations 3.1.6. Applications to Discontinuous O.D.Eqns 3.1.7. Bounded Solutions Close to Homoclinics 3.2. Bifurcation of Periodics from Homoclinics II 3.2.1. Singular Discontinuous O.D.Eqns 3.2.2. Linearized Equations 3.2.3. Bifurcation of Subharmonics 3.2.4. Applications to Singular Discontinuous O.D.Eqns 3.3. Bifurcation of Periodics from Periodics 3.3.1. Discontinuous O.D.Eqns 3.3.2. Linearized Problem 3.3.3. Bifurcation of Periodics in Nonautonomous Systems 3.3.4. Bifurcation of Periodics in Autonomous Systems 3.3.5. Applications to Discontinuous O.D.Eqns 3.3.6. Concluding Remarks 3.4. Bifurcation of Periodics in Relay Systems 3.4.1. Systems with Relay Hysteresis 3.4.2. Bifurcation of Periodics 3.4.3. Third-Order O.D.Eqns with Small Relay Hysteresis 3.5. Nonlinear Oscillators with Weak Couplings 3.5.1. Weakly Coupled Systems 3.5.2. Forced Oscillations from Single Periodics 3.5.3. Forced Oscillations from Families of Periodics 3.5.4. Applications to Weakly Coupled Nonlinear Oscillators 4. Bifurcation of Chaotic Solutions 4.1. Chaotic Differential Inclusions 4.1.1. Nonautonomous Discontinuous O.D.Eqns 4.1.2. The Linearized equation4.1.3. Bifurcation of Chaotic Solutions 4.1.4. Chaos from Homoclinic Manifolds 4.1.5. Almost and Quasi Periodic Discontinuous O.D.Eqns 4.2. Chaos in Periodic Differential Inclusions 4.2.1. Regular Periodic Perturbations 4.2.2. Singular Differential Inclusions 4.3. More about Homoclinic Bifurcations 4.3.1. Transversal Homoclinic Crossing Discontinuity 4.3.2. Homoclinic Sliding on Discontinuity 5. Topological Transversality 5.1. Topological Transversality and Chaos 5.1.1. Topologically Transversal Invariant Sets 5.1.2. Difference Boundary Value Problems 5.1.3. Chaotic Orbits 5.1.4. Periodic Points and Extensions on Invariant Compact Subsets 5.1.5. Perturbed Topological Transversality 5.2. Topological Transversality and Reversibility 5.2.1. Period Blow-up 5.2.2. Period Blow-up for Reversible Diffeomorphisms 5.2.3. Perturbed Period Blow-up 5.2.4. Perturbed Second Order O.D.Eqns5.3. Chains of Reversible Oscillators 5.3.1. Homoclinic Period Blow-up for Breathers5.3.2. Heteroclinic Period Blow-up for Non-Breathers5.3.3. Period Blow-up for Traveling Waves 6. Traveling waves on lattices 6.1. Traveling Waves in Discretized P.D.Eqns 6.2. Center Manifold Reduction 6.3. A Class of Singularly Perturbed O.D.Eqns6.4. Bifurcation of Periodic Solutions 6.5. Traveling Waves in Homoclinic Cases6.6. Traveling Waves in Heteroclinic Cases6.7. Traveling Waves in 2 Dimensions 7. Periodic Oscillations of Wave Equations7.1. Periodics of Undamped Beam Equations7.1.1. Undamped Forced Nonlinear Beam Equations7.1.2. Existence Results on Periodics 7.1.3. Subharmonics from Homoclinics7.1.4. Periodics from Periodics 7.1.5. Applications to Forced Nonlinear Beam Equations7.2. Weakly Nonlinear Wave Equations 7.2.1. Excluding Small Divisors 7.2.2. Lebesgue Measures of Nonresonances7.2.3. Forced Periodic Solutions 7.2.4. Theory of Numbers and Nonresonances8. Topological Degree for Wave Equations 8.1. Discontinuous Undamped Wave Equations8.2. Standard Classes of Multi-Mappings 8.3. M-Regular Multi-Functions 8.4. Classes of Admissible Mappings8.5. Semilinear Wave Equations 8.6. Construction of Topological Degree8.7. Local Bifurcations 8.8. Bifurcations from Infinity8.9. Bifurcations for Semilinear Wave Equations8.10. Chaos for Discontinuous Beam Equations Bibliography Index